Abstract

In this paper, we give a faster width-dependent algorithm for mixed packingcovering LPs. Mixed packing-covering LPs are fundamental to combinatorial optimization in computer science and operations research. Our algorithm finds a 1 + ε approximate solution in time where N is number of nonzero entries in the constraint matrix, and w is the maximum number of nonzeros in any constraint. This algorithm is faster than Nesterov’s smoothing algorithm which requires   time, where n is the dimension of the problem. Our work utilizes the framework of area convexity introduced in [Sherman-FOCS’17] to obtain the best dependence ε while breaking the infamous barrier to eliminate the factor of The current best width-independent algorithm for this problem runs in time [Young-arXiv-14] and hence has worse running time dependence on ε. Many real life instances of mixed packing-covering problems exhibit small width and for such cases, our algorithm can report higher precision results when compared to width-independent algorithms. As a special case of our result, we report a 1 + ε approximation algorithm for the densest subgraph problem which runs in time where m is the number of edges in the graph and d is the maximum graph degree.